Geodesic
On k-Path Pancyclic Graphs
Discussiones Mathematicae. Graph Theory, Tome 35 (2015) no. 2, pp. 271-281

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For integers k and n with 2 ≤ k ≤ n − 1, a graph G of order n is k-path pancyclic if every path P of order k in G lies on a cycle of every length from k + 1 to n. Thus a 2-path pancyclic graph is edge-pancyclic. In this paper, we present sufficient conditions for graphs to be k-path pancyclic. For a graph G of order n ≥ 3, we establish sharp lower bounds in terms of n and k for (a) the minimum degree of G, (b) the minimum degree-sum of nonadjacent vertices of G and (c) the size of G such that G is k-path pancyclic
Keywords: Hamiltonian, panconnected, pancyclic, path Hamiltonian, path pancyclic 
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Bi, Zhenming; Zhang, Ping. On k-Path Pancyclic Graphs. Discussiones Mathematicae. Graph Theory, Tome 35 (2015) no. 2, pp. 271-281. http://geodesic.mathdoc.fr/item/DMGT_2015_35_2_a6/